Rewrite as a simplified fraction. ${2.24\overline{7} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 2247.7777...\\ 100x &= 224.7777...\end{align*}}$ Subtracting the two equations, we have: ${900x = 2023}$ ${x = \dfrac{2023}{900}} \text{or} {2 \dfrac{223}{900}}$